{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:IWHUOWR6SDSASUOTTGLCIMDEPF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d2fd24b8c9b83bcc5bad730ab2f55f1e15d07db5ee63ec7f5304a1b72d49f7c6","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-10T03:53:40Z","title_canon_sha256":"07ee2b464abfd845f90b62657012b2b1b3dead455d7e6adda86adfd6f732d7c6"},"schema_version":"1.0","source":{"id":"1110.1921","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1110.1921","created_at":"2026-05-18T03:34:16Z"},{"alias_kind":"arxiv_version","alias_value":"1110.1921v4","created_at":"2026-05-18T03:34:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1921","created_at":"2026-05-18T03:34:16Z"},{"alias_kind":"pith_short_12","alias_value":"IWHUOWR6SDSA","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_16","alias_value":"IWHUOWR6SDSASUOT","created_at":"2026-05-18T12:26:32Z"},{"alias_kind":"pith_short_8","alias_value":"IWHUOWR6","created_at":"2026-05-18T12:26:32Z"}],"graph_snapshots":[{"event_id":"sha256:60a169a5f33d8fb5237cc5aa7083219ba8422005221a9dfb43f051663f8eec18","target":"graph","created_at":"2026-05-18T03:34:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this note, we investigate genera for the slopes of a knotted torus in the 4-sphere analogous to the genus of a classical knot. We compare various formulations of this notion, and use this notion to study the extendable subgroup of the mapping class group of the knotted torus.","authors_text":"Hongbin Sun, Shicheng Wang, Yi Liu, Yi Ni","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-10T03:53:40Z","title":"On slope genera of knotted tori in 4-space"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1921","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:89a2e0a89dfb1ab7c6b2f29ba9e8938d572022d5b257e5255a031a39ebb9b5a9","target":"record","created_at":"2026-05-18T03:34:16Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d2fd24b8c9b83bcc5bad730ab2f55f1e15d07db5ee63ec7f5304a1b72d49f7c6","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2011-10-10T03:53:40Z","title_canon_sha256":"07ee2b464abfd845f90b62657012b2b1b3dead455d7e6adda86adfd6f732d7c6"},"schema_version":"1.0","source":{"id":"1110.1921","kind":"arxiv","version":4}},"canonical_sha256":"458f475a3e90e40951d3999624306479764a8f47f1e13519f270412a929ee9c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"458f475a3e90e40951d3999624306479764a8f47f1e13519f270412a929ee9c0","first_computed_at":"2026-05-18T03:34:16.434638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:34:16.434638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dnJrwObMJIQpPaXctpb73ke1Yeo+89D2GFhFIQTaaQrtd8lkeTW0pHa18TQh3/yKyC6gl7rN9Mew8MVSpTaDCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:34:16.435555Z","signed_message":"canonical_sha256_bytes"},"source_id":"1110.1921","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:89a2e0a89dfb1ab7c6b2f29ba9e8938d572022d5b257e5255a031a39ebb9b5a9","sha256:60a169a5f33d8fb5237cc5aa7083219ba8422005221a9dfb43f051663f8eec18"],"state_sha256":"3186c08fadbbd635b3c02ecb40bcbbfa9a4f6e94649948bfac08bc3aaa7ff451"}